A fundamental solution for a linear partial differential equation with regard to the Dirac delta function is the solution of the inhomogeneous equation.

where

$G(\mathbf{x},\mathbf{y})$ is the three-dimensional fundamental solution for the Laplace equation,

$\mathbf{x}$ is the spatial coordinate which is collocated on the boundary,

$\mathbf{y}$ is the coordinate of source points, and

$\delta (\mathbf{x}-\mathbf{y})$ is the Dirac delta function. In the MFS, the unknown is assumed to be the linear combination of fundamental solutions of the governing equation using source points. The solution of the Laplace equation in three dimensions is approximated as follows:

where

${\alpha}_{j}$ is the coefficient or the intensity of source points,

${\mathbf{y}}_{j}$ is the source placed outside the domain, and

$N$ is the source number. The fundamental solution of three-dimensional Laplace equation is then expressed as

where

${r}_{j}=\left|\mathbf{x}-{\mathbf{y}}_{j}\right|$ is the distance between the

$\mathbf{x}$ and

j-th sources

${\mathbf{y}}_{j}$. Applying the boundary conditions, the following equations can be obtained:

where

$k=1,\dots ,Q$,

$Q$ is the boundary point number,

$g({\mathbf{x}}_{k})$ and

$f({\mathbf{x}}_{k})$ are the Dirichlet and Neumann boundary values given at boundary points, respectively. In order to determine the coefficients,

${\alpha}_{j}$, we may collocate the boundary collocation and source points using Equations (13) and (14). Then, the following simultaneous linear equations may be obtained as

where

$\mathbf{A}=\frac{1}{4\pi}\left[\begin{array}{cccccc}1/{r}_{11}& 1/{r}_{12}& 1/{r}_{13}& 1/{r}_{14}& \cdots & 1/{r}_{1N}\\ 1/{r}_{21}& 1/{r}_{22}& 1/{r}_{23}& 1/{r}_{24}& \cdots & 1/{r}_{2N}\\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ 1/{r}_{i1}& 1/{r}_{i2}& 1/{r}_{i3}& 1/{r}_{i4}& \cdots & 1/{r}_{iN}\\ {r}_{11}\xb7{\overrightarrow{n}}_{11}/{r}_{11}^{3}& {r}_{12}\xb7{\overrightarrow{n}}_{12}/{r}_{12}^{3}& {r}_{13}\xb7{\overrightarrow{n}}_{13}/{r}_{13}^{3}& {r}_{14}\xb7{\overrightarrow{n}}_{14}/{r}_{14}^{3}& \cdots & {r}_{1N}\xb7{\overrightarrow{n}}_{1N}/{r}_{1N}^{3}\\ {r}_{21}\xb7{\overrightarrow{n}}_{21}/{r}_{21}^{3}& {r}_{22}\xb7{n}_{22}/{r}_{22}^{3}& {r}_{23}\xb7{\overrightarrow{n}}_{23}/{r}_{23}^{3}& {r}_{24}\xb7{\overrightarrow{n}}_{24}/{r}_{24}^{3}& \cdots & {r}_{2N}\xb7{\overrightarrow{n}}_{2N}/{r}_{2N}^{3}\\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots \\ {r}_{j1}\xb7{\overrightarrow{n}}_{j1}/{r}_{j1}^{3}& {r}_{j2}\xb7{\overrightarrow{n}}_{j2}/{r}_{j2}^{3}& {r}_{j3}\xb7{\overrightarrow{n}}_{j3}/{r}_{j3}^{3}& {r}_{j4}\xb7{\overrightarrow{n}}_{j4}/{r}_{j4}^{3}& \cdots & {r}_{jN}\xb7{\overrightarrow{n}}_{jN}/{r}_{jN}^{3}\end{array}\right]$,

$\mathsf{\alpha}={\left[{\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}\right]}^{T}$,

$\mathbf{b}={\left[{g}_{1},{g}_{2},\dots ,{g}_{i},{f}_{1},{f}_{2},\dots ,{f}_{j}\right]}^{T}$. In the above equations,

$\mathbf{A}$ is a

$Q\times S$ matrix,

$S$ is the source number,

$\mathsf{\alpha}$ is a vector (size of

$S\times 1$) of unknown coefficients,

$\mathbf{b}$ is a vector (size of

$Q\times 1$) of given values from boundary conditions at collocation points.

$i$ and

$j$ are the boundary point number for Dirichlet and Neumann values, respectively,

${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$ are unknowns which need to be determined,

${r}_{11},{r}_{12},\dots ,{r}_{jN}$ are distances,

${\overrightarrow{n}}_{11},{\overrightarrow{n}}_{12},\dots ,{\overrightarrow{n}}_{jN}$ are outward normal directions,

${g}_{1},{g}_{2},\dots ,{g}_{i}$ and

${f}_{1},{f}_{2},\dots ,{f}_{j}$ are the boundary data.